The Tea Party has fallen. Now what?

FORTUNE — Tactically, the Republican establishment is routing the Tea Party. The insurgency’s backslide has been apparent all year, as its handpicked challengers to GOP incumbents failed to gain traction, groups representing it in Washington overreached, and the deficit concerns stoking its base waned. But yesterday, the “backslide” slid right back off a cliff. Tea Party-backed candidates in three key primary races suffered decisive losses in Kentucky, Georgia, and Idaho.

With the handwriting on the wall, deep-pocketed conservative sponsors huddled last Thursday and stewed over how to force the GOP to double down on hard-right policy positions. Those include opposition to a big immigration deal, same-sex marriage, and abortion rights — issues toxic to the imperative of broadening the party’s demographic coalition. But the movement’s electoral drubbing suggests its grip on the Republican agenda may finally be breaking.

The question is what will replace it. Senate Minority Leader Mitch McConnell, one of the big victors in yesterday’s contests, was explicit with Fortune earlier this year that Senate Republicans will not unify behind a governing vision before the November midterms. And even if a more moderate brand of Republicanism is ascendant, the term itself remains relative — and murky.

Consider that Thom Tillis, the establishment’s man in the North Carolina Senate primary, led the ultraconservative state House and as recently as 2011 talked about the need to “divide and conquer” those on public assistance. Or conversely that Ben Sasse, who delivered a rare victory for Tea Party groups in the Nebraska Senate primary by affecting an anti-establishment pose, was hardly an outsider, having gone to Harvard and served in George W. Bush’s administration.

Amidst this identity crisis, Republicans across the ideological spectrum are clamoring to lay claim to the legacy of a figure they are lately elevating to mythic status: Jack Kemp, the former Congressman and vice presidential candidate who died in 2009. It makes sense. As an animating force behind Ronald Reagan’s supply-side economic program, Kemp’s Reaganite credentials were ironclad. But he was also perhaps the last major GOP figure who invested in preaching compassion for the working poor while reaching out to minority constituencies not accustomed to hearing from Republicans. For Kemp, that outreach was a natural extension of his efforts on the field as an all-star NFL quarterback leading integrated teams. His would-be political heirs are having a somewhat toughertime repeating the trick.

In truth, Kemp reveled in his iconoclasm and would likely fare poorly in a conservative purity test today. His call for swift action to aid struggling homeowners during the mortgage crisis, for example, put him at odds with the new right’s founding cri de coeur. And he was a passionate advocate of immigration reform. No matter. Sasse in his victory speech quoted Kemp: “We may not get every vote, but we’ll speak to every heart.” And Sasse’s aides in a memo said the candidate will model himself after Kemp in office, following the late pol’s record of providing “real ideas and real solutions to real problems” — a subtle but unmistakable dig at the just-say-no style that’s come to characterize Texas Senator Ted Cruz and like-minded GOP reactionaries. Sasse was only the latest in a growing number of Republicans looking to borrow some legitimacy by association with Kemp.

So in a fitting counterpoint to last Thursday’s meeting of conservative backers, the Jack Kemp Foundation convened a small group of thinkers on Friday to chart some new policy priorities. While the right-wing meeting had taken place at the Ritz-Carlton in Tyson’s Corner, just outside Washington, the Kemp group met at Lincoln’s Cottage — the 16th president’s summer home — perched on a hill three miles north of the White House. These days, the Greek revival mini-mansion stands hidden on the grounds of a retirement home for veterans in the midst of a working-class neighborhood, a spare and largely forgotten Lincoln memorial in a city full of them.

Most of the Kemp conference unfolded in the Emancipation Room, the bedroom where Lincoln crafted the document that freed the slaves, securing a moral birthright for his fledgling party in the process. That may sound like a portentous place for a clutch of boldfaced names from the conservative establishment — Bill Kristol, Fred Barnes, Peggy Noonan, Amity Shlaes, etc. — to mull how to bind up the GOP’s wounds. But the issue of the party’s internal division somehow never came up directly. Instead, much of the morning was given over to familiar bellyaching about the wages of Obama’s foreign-policy fecklessness on American prestige abroad.

Only late in the afternoon did a participant spell out in stark terms the challenge facing the party. “People don’t think conservatives care,” Arthur Brooks, president of the American Enterprise Institute, told a dwindling crowd. “They think conservatives don’t care about the poor … You can’t win if people think you won’t be remotely sympathetic to people like them.”

While more aggressive institutions to its right have stumbled embracing dead-end extremism, AEI of late has been reasserting itself as a voice of right-leaning reasonableness, which should make it an interesting place to watch if the GOP establishment’s revival proves durable. What Brooks was offering Friday, for the most part, was advice on framing. “The biggest reason that people, particularly people who are poor, think that Republicans, conservatives, don’t care about people like them is because they talk as if they don’t,” he said.

Yet Brooks also suggested that to pull off a new sales pitch, Republicans first have to buy it themselves. “Jack Kemp — you know what his magic was? He loved people. He believed in them,” Brooks said. “People who are not working would much prefer to have their children see them earning a paycheck as opposed to getting a welfare check. That’s the truth. And if you believe the best about people and give them opportunity, if you give them hope, they will respond largely in a very constructive way. But if you assume the worst about them, then they will assume that you are hostile, they will assume you are racist, they’ll assume you don’t care about people like them, and to a certain extent they won’t be wrong.”

Republican maneuverings between now and Election Day will amount to getting the game pieces in place. Where the balance of power in the party will truly reside and what that means for the fate of the Republican renovation remain questions for a later day.